[Ether]

Using the analysis results from Gus' last post, and the hint in post 13, the final solution step is quite straightforward.

Open a LibreOffice spreadsheet and populate columns A and B with the XY data. Label the data in column A x_data. Label the data in column B y_data.

In column C put degrees from 139 to 199 stepping by 1.

In column D put radians equivalent of column C: colD=colC/180*pi()

Initialize cell O2 to zero

Label cell P2 "_b" and initialize it to one.

Label cell L2 "_a" and put the formula = O2/180*pi()

In column E put the 2-parameter polar equation model:
r = (colD - _a)^_b

In column G put formula colE*COS(colD)+1.16
Label the data cells in this column x_model

In column H put the formula colE*SIN(colD)-0.3
Label the data cells in this column y_model

In cell M2 put the formula SUMXMY2(x_data,x_model) + SUMXMY2(x_data,x_model). This is the objective function for the Solver.

Tools | Solver
Target Cell M2
Minimum
By changing cells O2 : P2
Solve
click continue a couple of times to refine the value